#303054

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,8)(3,7,4,6)(9,10,15,11)(12,14,16,13)(17,18,22,26)(19,27,30,24)(20,23,25,21)(28,32,31,29);(1,3,5,4)(2,6,8,7)(9,11,15,10)(12,13,16,14)(17,19)(18,24)(20,31)(21,29)(22,30)(23,32)(25,28)(26,27);(9,12,14,11,15,16,13,10)(17,20,23,26,22,25,21,18)(19,28,29,27,30,31,32,24);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,12,11,16)(17,21,22,23)(18,25,26,20)(19,29,30,32)(24,28,27,31)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=f,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210250);